Volume 18, issue 1 (2018)

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The closed-open string map for $S^1$–invariant Lagrangians

Dmitry Tonkonog

Algebraic & Geometric Topology 18 (2018) 15–68
Abstract

Given a monotone Lagrangian submanifold invariant under a loop of Hamiltonian diffeomorphisms, we compute a piece of the closed-open string map into the Hochschild cohomology of the Lagrangian which captures the homology class of the loop’s orbit.

Our applications include split-generation and nonformality results for real Lagrangians in projective spaces and other toric varieties; a particularly basic example is that the equatorial circle on the 2–sphere carries a nonformal Fukaya A algebra in characteristic 2.

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Keywords
circle action, Lagrangian submanifold, Floer homology, closed-open map, Fukaya category, split-generation, formality
Mathematical Subject Classification 2010
Primary: 53D37, 53D40, 57R58
Secondary: 53D45
References
Publication
Received: 8 April 2015
Revised: 19 May 2017
Accepted: 11 June 2017
Published: 10 January 2018
Authors
Dmitry Tonkonog
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge
United Kingdom